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This study presents Green's functions of parallel-plate structures, where one plate has a smooth conducting surface and the other an artificial surface realised by a one-dimensional or two-dimensional periodic metamaterial-type texture. The purpose of the periodic texture is to provide cut-off of the lowest order parallel-plate modes, thereby forcing electromagnetic energy to follow conducting ridges or strips, that is, to form a gap waveguide as recently introduced. The Green's functions are constructed by using the appropriate homogenised ideal or asymptotic boundary conditions in the plane-wave spectral domain, thereby avoiding the complexity of the Floquet-mode expansions. In the special case of a single ridge or strip, an additional numerical search for propagation constants is needed and performed in order to satisfy the boundary condition on the considered ridge or strip in the spatial domain. The results reveal the dispersion characteristics of the quasi-transverse electromagnetic modes that propagate along the ridges or strips, including their lower and upper cut-off frequencies, as well as the theoretical decay of the modal field in the transverse cut-off direction. This lateral decay shows values of 50-100 dB per wavelength for realisable geometries, indicating that the gap waveguide modes are extremely confined. The analytical formulas for the location of the stopband of the lowest order parallel-plate modes obtained by small-argument approximation of the dispersion equation are also shown. To verify the proposed analysis approach, the results are compared with the results obtained with a general electromagnetic solver showing very good agreement.